Topic3NORMAL1

Now that you know what is the mean and standard deviation of a set of scores, we can examine the concept of a normal distribution. The normal curve was developed mathematically in 1733 by DeMoivre as an appropximation to the binomial distribution. Laplace used the normal curve in 1783 to describe the distribution of errors. However, is was Gauss who popularised the normal curve when he used it to analyse astronomical data in 1809 and it became known as the Gaussian distribution. The term normal distribution refers to a particular way in which scores or observations will tend to pile up or distribute around a particular value rather than be scattered all over (See graph below). The normal distribution which is bell-shaped is based on a mathematical equation (which we will not get into!).

While some argue that in the real world, scores or observations are seldom normally distributed, others argue that in the general population, many variables such as height, weight, IQ scores, reading ability, job satisfaction, blood pressure turn out to have distributions that are bell-shaped or normal.

The normal distribution is important for TWO reasons:

Many physical, biological and social phenomena or variables are normally distributed. However, some variables are only approximately normally distributed.

Many kinds of statistical tests (such as the t-test, ANOVA) are derived from a normal distribution. In other words, most of these statistical tests argue that they work best when the sample tested is distributed normally.

FORTUNATELY, these statistical tests work very well even if the distribution is only approximately normally distributed. Some tests work well even with very wide deviations from normality. They are described as 'robust' tests that are able to tolerate the lack of a normal distribution.

EXAMPLE:

The graph above is a picture of a normal distribution of IQ scores among a sample of 16 year old adolescents. The mean is 100 and the standard deviation is 15. As you can see, the distribution is symmetric. If you folded the graph in the centre, the two sides would match, i.e. they are identical. The centre of the distribution is the mean. The mean of a normal distribution is also the most frequently occuring value (i.e. the mode), and it is also the value that divides the distribution of scores into two equal parts (i.e. the median). In any normal distribution, the mean, median and the mode all have the same value (i.e. 100 in the example above).

The normal distribution shows the area under the curve. See the graph above and you will notice that with a mean of 100 and a standard deviation of 15; 68% of all IQ scores fall between 85 (i.e. one standard deviation less then the mean which is 100 - 15 = 85) and 115. (i.e. one standard deviation more than the mean which is 100 + 15 = 115). A normal distribution can have any mean and standard deviation. But the percentage of cases or individuals falling within one, two or three standard deviations from the mean is always the same. The shape of a normal distribution does not change. Means and standard deviations will differe from variable to variable. But the percentage of cases or individuals falling within specific intervals is alwasys the same in a true normal distribution.